If a and b are integers such that a - b = 100, find the minimum value of a*b.
The closer two numbers with the same sum are, the bigger their product.
In this case, the problem did not specify whether positive or negative so the minimum value is: 50-(-50)=100
Therefore, a*b= -2500
Because \((a,b )\in \mathbb Z^2\), it is optimal to minimize b and maximize a.
Because of that reason, if we choose (a, b) = (50, -50), then we get our minimum value, which is -2500.
(If a was larger, then b would be smaller. Same otherwise. It is actually optimal to choose |a| = |b|, or make |a| as close as |b|.)